Complex Numbers and Quadratic Equations · Question 3

JEE Advanced 2025 Paper 1MCQMedium📈 Topic in 33% papers+4−1

Let R denote the set of all real numbers. Let ai, bi R for i \1, 2, 3\. Define the functions f: R R, g: R R, and h: R R by f(x) = a1 + 10x + a2x2 + a3x3 + x4, g(x) = b1 + 3x + b2x2 + b3x3 + x4, h(x) = f(x+1) - g(x+2). If f(x) g(x) for every x R, then the coefficient of x3 in h(x) is

8
2
-4
-6

Answer

(3) -4

Solution

Key idea: a real polynomial of odd degree always has a real root.

f(x) - g(x) = (a1 - b1) + 7x + (a2 - b2)x2 + (a3 - b3)x3 must have no real root, so it cannot be a cubic: a3 = b3.

Coefficient of x3 in f(x+1): from (x+1)4 it is 4, from a3(x+1)3 it is a3; total 4 + a3.

Coefficient of x3 in g(x+2): from (x+2)4 it is 41 2 = 8, plus b3; total 8 + b3.

Coefficient of x3 in h(x) = (4 + a3) - (8 + b3) = -4 (as a3 = b3).

Answer: (C)

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Let R denote the set of all real numbers. Let z1 = 1 + 2i and z2 = 3i be two complex numbers, where i = -1. Let S = \(x, y) R R : |x + iy - z1| = 2|x + iy - z2|\. Then which of the following statements is (are) TRUE?

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